MASTER Recherche de Mathématiques Université Bordeaux I UFR ...

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MASTER Recherche de Mathématiques Université Bordeaux I UFR ...

Publié le : jeudi 21 juillet 2011
Lecture(s) : 108
Nombre de pages : 11
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MASTERRecherchedeMathe´matiques ParcoursSp´ecialise´ Universite´BordeauxI U.F.R.Mathe´matiquesetInformatique Anne´e2006/07
Programme des cours
Pour plus d’information, contacter le responsableemmanuel.kowalski@math.u-bordeaux1.fr. Pourlinscription,contacterlesecre´tariatchristine.parison@math.u-bordeaux1.fr LesiteinternetduParcoursSp´ecialis´eesta`ladressehttp://www.math.u-bordeaux1.fr/ emi/Master/MM2S/MM2S.html Pour plus de renseignement concernant le Master ALGANT, dans le cadre duquel une partie des cours ont lieu, voirhttp://www.math.u-bordeaux1.fr/ALGANT
The´oriedesNombres CoursMAT912dupremiersemestre
Number theory Ph.Cassou-Nogu`es
Re´sum´e.snouNrecnniatpoleureptioulsmbnodreossndniuopsoropetded´evtroduire classiquesmaiscruciauxpourle´tudedelath´eoriedesnombres.Nousnousint´eressonsen particulierauxcorpslocauxet`alaramicationdeleursextensions.Nousdonnonslesr´esultats delathe´orieducorpsdeclasseslocaletglobaletnousde´montronsquelquesconse´quencesde cesr´esultats. Plan : – Corps finis – Corps locaux et valuations. – Ramification des extensions d’un corps local. Introduction`alathe´orieducorpsdeclasses.
Abstract.The goal of this course is to introduce and develop various classical tools which play a crucial role in Number Theory. In particular we study local fields and the ramification of their extensions. Moreover we state the main results of local and global class field theory and prove some of their consequences.
R´efe´rences/Bibliography. [CF]J.W.S.Cassels,A.Fr¨ohlich:Algebraic Number Theory, Academic Press, 1967. [FT]A.Fro¨hlich,M.J.Taylor:Algebraic Number Theory, Cambridge University Press 27, 1991. [G] G. Gras :Class field theory, from theory to practice, Springer Verlag, 2003 [KKS] K. Kato, N. Kurokawa, T. Saito :Number Theory 1. Fermat’s Dream, Translations of Mathematical monographs 186, 2000. [M] S. Milne :Algebraic Number Theory, page Web de Milne. [S] J.P. Serre :A course in Arithmetic, Corrected fifth printing, Graduate Texts in Mathe-matics 7, Springer Verlag, 1996. [S] J.P. SerreCorps locaux, Hermann, 1968 [Sw] H. P. F. Swinnerton-Dyer :A brief guide to Algebraic Number Theory, Cambridge Uni-versity Press, 2001. 2
Th´eoriealgorithmiquedesnombresavanc´ee CoursMAT913dupremiersemestre
Advanced computational number theory Karim Belabas
R´esume´.Le cours utilisera comme fil rouge les algorithmes classiques et modernes de factori-sationpourpre´senterdesid´eesettechniquesimportantesdeth´eoriealgorithmiquedesnombres. Ondiscuteradelar´eductiondesZednoitasmoˆnylopel,duxearitoacafelesomud´rsedtsees-en une variable sur un coprs fini, sur les rationnels et surC, puis on parler de tests de pri-malit´e(jusqua`lalgorithmedeprimalitybase´surlescourbeselliptiques),etdalgorithmesde factorisation(jusquaucriblealge´brique).Pendanttoutlecours,laccentseramissurlesid´ees importantes,paroppositionaveclesde´tailstechniques,n´ecessairespourdesimple´mentations ecaces,etaveclesme´thodesasymptotiquementrapides.
Abstract.The course uses classical and modern factorization algorithms to present impor-tant ideas and techniques in computational number theory. We will cover the reduction of Z-modules and lattices, factorization of univariate polynomials over finite fields, the rationals and the complex numbers, then primality testing (up to the Elliptic Curve Primality Proving algorithm) and integer factorization (up to the Number Field Sieve). The emphasis is on im-portant ideas throughout, as opposed to technical details necessary for efficient implementation, and asymptotically fast methods. Prerequisites.In the last part, basic facts about elliptic curves (overCand a finite field) and algebraic number theory (splitting of primes, class groups) will be sketched then assumed.
Bibliography. JoachimvonzurGathenandJu¨rgenGerhard,Modern computer algebra, Cambridge Univer-sity Press, New York, 1999. H. Cohen,A course in computational algebraic number theory, Springer-Verlag, 1996. Last year’s course notes are available athttp://www.math.u-bordeaux.fr/~belabas/teach/ 2005/MAT913/ 3
Introduction`ala CoursMAT914
g´eom´etriealge´briques dupremiersemestre
Introduction to algebraic geometry Q. Liu
Re´sum´e.unstnteicoCeseurla`ne´gaudoroitcseec´anavusebrialg´trieom´esnlptooieLnsuq.e sontillustre´esparlescourbesalge´briques. Plan : -Comple´mentsdalg`ebrecommutative. -Ensemblesalge´briques,vari´et´esanes,sche´masanes. -Faisceaux,sch´emas,varie´t´esalge´briques,vari´et´esprojectives. -Propri´ete´stopologiquesdesche´mas. - Quelques classes de morphismes. -Proprie´t´eslocales,normalisation. -Espacetangent,sche´masre´guliers,crit`erejacobien. -Calculdi´erentielles. - Diviseurs, faisceaux inversibles et groupe de Picard. -Courbesalge´briques.The´ore`medeRiemann-Roch,Th´eor`emedeRiemann-Hurwitz.
Abstract.This course is an introduction to algebraic geometry. It ends with Riemann-Roch theorem for algebraic curves. - Some topics on commutative algebra. - Algebraic sets, affine varieties and affines schemes. - Sheaves, schemes, algebraic varieties and projective varieties. - Tological proprieties of schemes. - Some classes of morphisms. - Local properties, normalization. - Tangent space, regular schemes, Jacobien criterion. - Differentials. - Divisors, invertible sheaves and Picard groups. - Algebraic curves. Riemann-Roch Theorem, Riemann-Hurwitz Theorem.
R´ef´erences/Bibliography. [H] R. Hartshorne :Algebraic geometry, Springer (1977). [L] A. Liu :Algebraic geometry and arithmetic curves, Oxford Univ. Press (2002). [M] D. Mumford :The Red Book of Varieties and Schemes, Lect. in Maths.1358, Springer (1988). [S] I. Shafarevich :Basic algebraic geometry, Springer (1994). 4
Equationsdevolutionetleurcontroˆle dans les espaces de Banach CoursMAT915dupremiersemestre
Evolution Equations and their control theory in Banach spaces E.M. Ouhabaz
R´esum´e.esbjectifdLotigaslirapenudesrsouece;bloutdsl´nuoitasriceser´estdeprlesente `al´etudedese´quationsdevolutiondanslesespacesdeBanachet,dautrepart,dintroduire certainesme´thodesettechniquesdelathe´orieducontroˆle.Danslapremi`erepartie,one´tudiera, entreautre,lesope´rateursnon-born´es,the´oriespectrale,re´solvante,semi-groupesdop´erateurs, th´eore`mesdege´ne´rationdesemi-groupes,stabilite´etde´croissanceexponentielle,op´erateurs auto-adjoints...Toutceciseraillustr´epardi´erentsexemples,notammentle´quationdesondes etles´equationsparaboliquesassocie´es`adesope´rateursdi´erentielssoumisa`diversesconditions d aux bord (Dirichlet, Neumann, mixtes...) dans les ouverts deR.Dans la seconde partie, on e´tudieradesquestionsdecontroˆlabilit´eetobservabilit´edes´equationsdevolution.Ilsagiraen grosducontˆoleapproximatif,exacte,conditiondeKalman,me´thodesHUMetGCC. Cecoursfourniratouteslesbasesn´ecessairespourpoursuivrelecoursdAnalysedusecond semestre.
Abstract.This course has two principal objectives, first of all to present the tools we need to study evolution equations in Banach spaces, then to introduce certain methods and techniques from control theory. The topics related to the evolution equations include: unbounded opera-tors, spectral theory, resolvants, semi-groups of operators, generation of semi-groups, stability and exponential decay of semi-groups, and self-adjoint operators. We will begin by defining and giving examples of unbounded operators and semi-groups, evaluating stability and decay , de-termining self-adjointness and finding out about the spectrum.We shall give many examples and illustrations including wave equations and parabolic equations associated with differential op-d erators and a variety of boundary conditions on open sets inR(Dirichlet, Neumann, mixed...) In the second part of the course we consider questions of controlability and observability for evolution equations, more precisely - approximative control, exact control, Kalman’s conditions, the HUM and GCC methods. This course gives all necessary preparation for the second semester analysis course.
Re´fe´rences/Bibliography. H. Brezis :Analyse Fonctionnelle, Masson 1984 J.L. Lions :rtnoaloˆCctxaPee.libieet´noesstattrruabitondesystbilisatiissdme`eesu´ibtr.loV. 1Masson 1988. V. Komornik, P. Loreti :Fourier series in control theory. Springer 2006. E.M. Ouhabaz :Analysis of Heat Equations on domains, Princeton Univ. Press 2004. J. Zabczyk :Mathematical control theory - An introduction, Springer-Verlag 1995. 5
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